Editing Constructible number (section)

==Equivalence of algebraic and geometric definitions==
If <math>a</math> and <math>b</math> are the non-zero lengths of geometrically constructed segments then elementary compass and straightedge constructions can be used to obtain constructed segments of lengths <math>a+b</math>, <math>|a-b|</math>, <math>ab</math>, and <math>a/b</math>. The latter two can be done with a construction based on the [[intercept theorem]]. A slightly less elementary construction using these tools is based on the [[geometric mean theorem]] and will construct a segment of length <math>\sqrt{a}</math> from a constructed segment of length <math>a</math>. It follows that every algebraically constructible number is geometrically constructible, by using these techniques to translate a formula for the number into a construction for the number.<ref>{{harvp|Herstein|1986|pp=236–237}}; {{harvp|Moise|1974|p=224}}; {{harvp|Fraleigh|1994|pp=426–427}}; {{harvp|Courant|Robbins|1996|pp=120–122|loc=Section III.1.1: Construction of fields and square root extraction}}.</ref>

{{multiple image|total_width=720|align=center|header=Compass and straightedge constructions for constructible numbers
|image1=Number construction multiplication.svg|caption1=<math>ab</math> based on the [[intercept theorem]]
|image2=Number construction division.svg|caption2=<math>\frac{a}{b} </math> based on the [[intercept theorem]]
|image3=Root_construction_geometric_mean5.svg|caption3=<math>\sqrt{p}</math> based on the [[geometric mean theorem#Constructing_a_square_root|geometric mean theorem]]}}

In the other direction, a set of geometric objects may be specified by algebraically constructible real numbers: coordinates for points, slope and <math>y</math>-intercept for lines, and center and radius for circles. It is possible (but tedious) to develop formulas in terms of these values, using only arithmetic and square roots, for each additional object that might be added in a single step of a compass-and-straightedge construction. It follows from these formulas that every geometrically constructible number is algebraically constructible.{{sfnmp|1a1=Martin|1y=1998|1pp=38–39|2a1=Courant|2a2=Robbins|2y=1996|2pp=131–132}}